Lepton flavor violation and scotogenic Majorana neutrino mass in a Stueckelberg $U(1)_{X}$ model

Apart from the Yukawa couplings, the most important effect to generate the Majorana mass from the scotogenic mechanism is the appearance of LNV term in the scalar potential, which dictates the scalar masses, couplings, and mixings. To examine these effects in the model, we write the scalar potential for the SM Higgs $H$ and $\eta_{1,2}$ as:

$\displaystyle\begin{split}V=&V_{\rm SM}+V(H,\eta_{1},\eta_{2})~{},\\ V_{\rm SM}=&-\mu^{2}_{H}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}~{},\\ V(H,\eta_{1},\eta_{2})=&\mu^{2}_{1}\eta^{\dagger}_{1}\eta_{1}+\mu^{2}_{2}\eta^{\dagger}_{2}\eta_{2}+\lambda_{1}(\eta^{\dagger}_{1}\eta_{1})^{2}+\lambda_{2}(\eta^{\dagger}_{2}\eta_{2})^{2}+\lambda_{3}(\eta^{\dagger}_{1}\eta_{1})(\eta^{\dagger}_{2}\eta_{2})\\ &+\lambda_{4}(\eta^{\dagger}_{1}\eta_{2})(\eta^{\dagger}_{2}\eta_{1})+\left(\lambda_{5}(H^{\dagger}\eta_{1})(H^{\dagger}\eta_{2})+\mbox{H.c.}\right)+\lambda_{6}(H^{\dagger}\eta_{1})(\eta^{\dagger}_{1}H)\\ &+\lambda_{7}(H^{\dagger}\eta_{2})(\eta^{\dagger}_{2}H)+\lambda_{8}(H^{\dagger}H)(\eta^{\dagger}_{1}\eta_{1})+\lambda_{9}(H^{\dagger}H)(\eta^{\dagger}_{2}\eta_{2})~{}.\end{split}$

(1)

It can be seen that the only non-self-Hermitian term comes from the $\lambda_{5}$ term, which violates the lepton number by two units and plays an important role on the radiative generation of the Majorana neutrino mass. The tiny neutrino mass can be achieved when $\lambda_{5}\ll 1$, which is the same as that shown in the Ma model Ma:2006km . For spontaneously breaking the electroweak gauge symmetry, we take $\mu^{2}_{H},\lambda_{H}>0$ as in the SM. The masses of $\eta_{1,2}$ can be irrelevant to the electroweak symmetry breaking, and we thus require $\mu^{2}_{1,2}(\lambda_{1,2})>0$. To preserve the $U(1)_{X}$ and $R_{p}$ symmetries, the vacuum expectation values (VEVs) of $\eta_{1,2}$ have to vanish. We therefore parametrize the components of the three doublet scalars as:

$$H=\left(\begin{array}[]{c}G^{+}\\ \frac{1}{\sqrt{2}}(v+h+iG^{0})\\ \end{array}\right)~{},~{}~{}\eta_{j}=\left(\begin{array}[]{c}\eta^{+}_{j}\\ \frac{1}{\sqrt{2}}(s_{j}+ia_{j})\\ \end{array}\right)~{},$$

(2)

where $G^{\pm,0}$ are the Goldstone bosons, $v$ is the VEV of $H$, and $h$ is the SM Higgs boson.

Since $\eta_{1}$ and $\eta_{2}$ carry different $U(1)_{X}$ charges, the charged scalars $\eta^{\pm}_{1}$ and $\eta^{\pm}_{2}$ do not mix and their masses are respectively obtained as:

$\displaystyle\begin{split}m^{2}_{\eta^{\pm}_{1}}&=\mu^{2}_{1}+\frac{\lambda_{8}v^{2}}{2}~{},\\ m^{2}_{\eta^{\pm}_{2}}&=\mu^{2}_{2}+\frac{\lambda_{9}v^{2}}{2}~{}.\end{split}$

(3)

Unlike the charged scalars, the neutral components of $\eta_{1,2}$ can mix via the $\lambda_{5}$ term. According to the scalar potential with Eq. (2), the mass-square matrices for $(s_{1},s_{2})$ and $(a_{1},a_{2})$ are respectively:

$$m^{2}_{S}=\left(\begin{array}[]{cc}m^{2}_{s_{1}}&m^{2}_{12}\\ m^{2}_{12}&m^{2}_{s_{2}}\\ \end{array}\right)~{},~{}~{}m^{2}_{A}=\left(\begin{array}[]{cc}m^{2}_{s_{1}}&-m^{2}_{12}\\ -m^{2}_{12}&m^{2}_{s_{2}}\\ \end{array}\right)~{},$$

(4)

with

	$\displaystyle m^{2}_{s_{1}}$	$\displaystyle=\mu^{2}_{1}+\frac{v^{2}}{2}(\lambda_{6}+\lambda_{8})~{},$
	$\displaystyle m^{2}_{s_{2}}$	$\displaystyle=\mu^{2}_{2}+\frac{v^{2}}{2}\left(\lambda_{7}+\lambda_{9}\right)~{},$
	$\displaystyle m^{2}_{12}$	$\displaystyle=\frac{v^{2}}{2}\lambda_{5}~{}.$		(5)

Each of the two $2\times 2$ mass-square matrices can be diagonalized by the corresponding orthogonal matrix $O(\theta_{\xi})$ ($\xi=S,A$) through $O(\theta_{\xi})m^{2}_{\xi}O^{T}(\theta_{\xi})$, where the $O(\theta_{\xi})$ matrix can be parametrized as:

$$O(\theta_{\xi})=\left(\begin{array}[]{cc}\cos\theta_{\xi}&\sin\theta_{\xi}\\ -\sin\theta_{\xi}&\cos\theta_{\xi}\\ \end{array}\right)~{}.$$

(6)

Since the matrix elements in $m^{2}_{A}$ are the same as those in $m^{2}_{S}$ except the sign change in the off-diagonal elements, we therefore take $\theta_{S}=-\theta_{A}\equiv\theta$. The eigenvalues for $m^{2}_{S}$ are found to be:

$$m^{2}_{S_{1,2}}=\frac{1}{2}\left[m^{2}_{s_{1}}+m^{2}_{s_{2}}\pm\sqrt{(m^{2}_{s_{2}}-m^{2}_{s_{1}})^{2}-4(m^{4}_{12})}\right]~{}.$$

(7)

For the physical pseudoscalars $A_{1,2}$, we have $m^{2}_{A_{1(2)}}=m^{2}_{S_{1(2)}}$. The mixing angle $\theta$ is given by:

$$\sin(2\theta)=-\frac{\lambda_{5}v^{2}}{m^{2}_{S_{2}}-m^{2}_{S_{1}}}~{}.$$

(8)

Because $S_{i}$ and $A_{i}$ are degenerate in mass, if one of them is the DM candidate, the large gauge interaction $A_{i}-S_{i}-Z$ will render too large a DM-nucleon scattering cross section. Hence, the possibility of using a scalar as the DM candidate in this model is excluded by the direct detection experiments. Instead, the singlet vector-like fermion $N$ becomes a promising DM candidate in the model. Since $\lambda_{5}$ term violates the lepton number and eventually leads to the Majorana mass, its value has to be sufficiently small, $\lambda_{5}\ll 1$ Ma:2006km , as alluded to earlier. As a result, the off-diagonal mass matrix element $|m_{12}^{2}|$ is suppressed and the mixing angle $\theta\ll 1$. In order to make the Yukawa couplings sufficiently large so that the LFV processes can be possibly detectable in the ongoing and planning experiments, we follow the approach in Toma:2013zsa and take $\lambda_{5}=10^{-9}$.

According to the $U(1)_{X}$ charges listed in Table 1, the Yukawa couplings of $\eta_{1,2}$ and $N$ to the SM leptons are given by:

$\displaystyle-{\cal L}_{Y}$

$\displaystyle=\overline{L}\,{\bf y}_{1}\,\tilde{\eta}_{1}N_{R}+\overline{L}\,{\bf y}_{2}\,\tilde{\eta}_{2}N^{C}_{L}+m_{N}\overline{N}_{L}N_{R}+\mbox{H.c.}~{},$

(9)

where the flavor indices are suppressed, $L^{T}=(\nu_{\ell},\ell)$ denotes the left-handed lepton doublet in the SM, ${\tilde{\eta}_{j}}=i\tau_{2}\eta^{*}_{j}$ with $\tau_{2}$ being a Pauli matrix, $N^{C}=C\gamma_{0}N^{*}$ is the charge conjugation of $N$, and $m_{N}$ is the mass of $N$. Although ${\bf y}_{1,2}$ can be generally complex, we can rotate away the phase of one of them by redefining the phases of the complex lepton doublet $L$. In our following analysis, we take the convention that ${\bf y}_{1}$ is real and ${\bf y}_{2}$ is complex, as parametrized by $y_{2k}=|y_{2k}|e^{i\phi_{k}}$ ($k=1,2,3$). Using Eqs. (2) and $(\ref{eq:omatrix})$, the Yukawa couplings in Eq. (9) can be decomposed as:

$\displaystyle\begin{split}-{\cal L}_{Y}=&-\overline{\ell}_{L}{\bf y}_{1}N_{R}\eta^{-}_{1}-\overline{\ell}_{L}{\bf y}_{2}N^{C}_{L}\eta^{-}_{2}+m_{N}\overline{N}_{L}N_{R}\\ &+\frac{1}{\sqrt{2}}\overline{\nu}_{\ell L}{\bf y}_{1}N_{R}\left[c_{\theta}(S_{1}-iA_{1})-s_{\theta}(S_{2}+iA_{2})\right]\\ &+\frac{1}{\sqrt{2}}\overline{\nu}_{\ell L}{\bf y}_{2}N^{C}_{L}\left[s_{\theta}(S_{1}+iA_{1})+c_{\theta}(S_{2}-iA_{2})\right]+\mbox{H.c.}~{},\end{split}$

(10)

where we denote $c_{\theta}~{}(s_{\theta})\equiv\cos\theta~{}(\sin\theta)$. We note that if we take the charged lepton $\ell_{L}$ as the mass eigenstate, in general, the $\eta^{\pm}_{1,2}$ Yukawa couplings are modified as $V^{\ell}_{L}{\bf y}_{i}$, where $V^{\ell}_{L}$ is a unitary matrix introduced for diagonalizing the charged lepton mass matrix. Therefore, the Yukawa couplings of $\eta^{\pm}_{i}$ are generally different from those of $S_{i}$ and $A_{i}$. Since we know nothing about the information of $V^{\ell}_{L}$ in the SM, we adopt $V^{\ell}_{L}=1$ to reduce the number of free parameters in the study. As a result, the effects contributing to the neutrino mass through the Yukawa couplings of $S_{i}$ and $A_{i}$ now have a strong correlation to the LFV processes that are mediated by $\eta^{\pm}_{i}$.

The Lagrangian invariant under the $SU(2)_{L}\times U(1)_{Y}\times U(1)_{X}$ gauge symmetry with the Stueckelberg gauge field included is given by:

$\displaystyle\begin{split}{\cal L}_{\rm kin}=&-\frac{1}{4}\vec{W}_{\mu\nu}\cdot\vec{W}^{\mu\nu}-\frac{1}{4}\hat{B}_{\mu\nu}\hat{B}^{\mu\nu}-\frac{1}{4}\hat{Z^{\prime}}_{\mu\nu}\hat{Z^{\prime}}^{\mu\nu}+\frac{m^{2}_{Z^{\prime}}}{2}\left(\hat{Z^{\prime}}^{\mu}-\frac{1}{m_{Z^{\prime}}}\partial^{\mu}B\right)^{2}\\ &-\frac{1}{2\beta}\left(\partial_{\mu}\hat{Z^{\prime}}^{\mu}+m_{Z^{\prime}}\beta B\right)^{2}+\overline{N}i\not{D}N+(D_{\mu}\eta_{1})^{\dagger}(D^{\mu}\eta_{1})+(D_{\mu}\eta_{2})^{\dagger}(D^{\mu}\eta_{2})~{},\end{split}$

(11)

where $W^{i}_{\mu\nu}$, $\hat{B}_{\mu\nu}$, and $\hat{Z^{\prime}}_{\mu\nu}$ are the gauge field stress tensors associated with the $SU(2)_{L}$, $U(1)_{Y}$, and $U(1)_{X}$ gauge groups, respectively, $B$ field is the Stueckelberg scalar field, $\beta$ is the gauge fixing parameter, and $m_{Z^{\prime}}$ is the mass of $U(1)_{X}$ Stueckelberg gauge field. We note that the kinetic mixing term $\hat{B}_{\mu\nu}\hat{Z^{\prime}}^{\mu\nu}$ can be rotated away by redefining the gauge fields $\hat{B}_{\mu}$ and $\hat{Z^{\prime}}_{\mu}$. However, we will show later that the mixings in $\gamma$-$Z^{\prime}$ and $Z$-$Z^{\prime}$ can be induced through radiative corrections.

The covariant derivatives on $N$ and $\eta_{i}$ are expressed as:

$\displaystyle\begin{split}D_{\mu}N&=(\partial_{\mu}+ig_{X}q_{N}\hat{Z^{\prime}}_{\mu})N~{},\\ D_{\mu}\eta_{i}&=\left(\partial_{\mu}+i\frac{g}{2}\vec{\tau}\cdot\vec{W}_{\mu}+i\frac{g^{\prime}}{2}\hat{B}_{\mu}+ig_{X}q_{\eta_{i}}\hat{Z^{\prime}}_{\mu}\right)\eta_{i}~{},\end{split}$

(12)

where $q_{N(\eta_{i})}$ denotes the $U(1)_{X}$ charge of $N(\eta_{i})$, and $g$ and $g^{\prime}$ are the gauge couplings of $SU(2)_{L}$ and $U(1)_{Y}$, respectively. The $NNZ^{\prime}$ interaction can be immediately obtained as:

$${\cal L}_{NNZ^{\prime}}=-g_{X}q_{N}\overline{N}\gamma^{\mu}N\hat{Z^{\prime}}_{\mu}~{}.$$

(13)

Although the quartic gauge couplings of $\eta_{i}$ to gauge bosons $W^{\pm}$, $Z$, $\gamma$, and $\hat{Z^{\prime}}$ can be derived from the kinetic terms of $\eta_{i}$, these couplings are irrelevant to our study here and are not presented explicitly. The various trilinear gauge couplings of $\eta_{i}$ from $(D_{\mu}\eta_{i})^{\dagger}(D^{\mu}\eta_{i})$ can be summarized as:

$\displaystyle\begin{split}{\cal L}_{\rm kin}\supset&~{}i\frac{g}{\sqrt{2}}\left\{\left[\partial_{\mu}\eta^{-}_{i}(S_{i}+iA_{i})-\eta^{-}_{i}(\partial_{\mu}S_{i}+i\partial_{\mu}A_{i})\right]W^{+\mu}+\mbox{H.c.}\right\}\\ &+\left(\frac{g}{2c_{W}}Z^{\mu}-g_{X}q_{X}\hat{Z^{\prime}}^{\mu}\right)\left[(\partial_{\mu}S_{i})A_{i}-S_{i}\partial_{\mu}A_{i}\right]\\ &+i\left(eA^{\mu}+\frac{1-2s^{2}_{W}}{2c_{W}}gZ^{\mu}+g_{X}q_{X}\hat{Z^{\prime}}^{\mu}\right)\left[(\partial_{\mu}\eta^{-}_{i})\eta^{+}_{i}-\eta^{-}_{i}\partial_{\mu}\eta^{+}_{i}\right]~{}.\end{split}$

(14)

The $W^{\pm}$, $Z$, and $\gamma$ gauge bosons and the Weinberg angle $\theta_{W}$ are defined through the relations:

$\displaystyle\begin{split}W^{\pm}&=\frac{1}{\sqrt{2}}(W^{1}_{\mu}\mp W^{2}_{\mu})~{},\\ A_{\mu}&=c_{W}\hat{B}_{\mu}+s_{W}W^{3}_{\mu}~{},\\ Z_{\mu}&=-s_{W}\hat{B}_{\mu}+c_{W}W^{3}_{\mu}~{},\end{split}$

(15)

where $c_{W}~{}(s_{W})\equiv\cos\theta_{W}~{}(\sin\theta_{W})$ and $e=gs_{W}=g^{\prime}c_{W}$ are used.

In the study of LFV processes, we need to know the photon and $Z$-boson gauge couplings to the SM particles. Therefore, the relevant photon and $Z$ gauge couplings are given by:

$${\cal L}_{Z}=-eQ_{f}\overline{f}\gamma_{\mu}fA^{\mu}-\frac{g}{2c_{W}}\overline{f}\left(C^{f}_{R}P_{R}+C^{f}_{L}P_{L}\right)\gamma_{\mu}fZ^{\mu}~{},$$

(16)

with

$$C^{f}_{R}=-2Q_{f}s^{2}_{W}~{},~{}~{}C^{f}_{L}=2T^{3}_{f}-2Q_{f}s^{2}_{W}~{},$$

(17)

where $Q_{f}$ is the electric charge of the fermion $f$, and $T^{3}_{f}$ is its third component of the weak isospin.

To obtain the Majorana neutrino mass in the model, we need both Yukawa couplings ${\bf y}_{1}$ and ${\bf y}_{2}$, which can ensure that the structure of Majorana mass term $L^{T}CL$ can be achieved. In addition, since $\eta_{1,2}$ have no VEVs and the lepton number is preserved in the ground state, we are left with the choice of using the lepton number-violating interaction $\lambda_{5}(H^{\dagger}\eta_{1})(H^{\dagger}\eta_{2})$ to generate the Weinberg’s dimension-5 operator $LHHL$ Weinberg:1979sa . For the purpose of clarity, we show a representative one-loop Feynman diagram in Fig. 1. According to the Yukawa couplings in Eq. (10), the neutrino mass matrix elements can be obtained as:

$$m^{\nu}_{ij}=\frac{\sin(2\theta)}{16\pi^{2}}\overline{y}_{ij}m_{N}\left[\frac{m^{2}_{S_{1}}}{m^{2}_{S_{1}}-m^{2}_{N}}\ln\left(\frac{m^{2}_{S_{1}}}{m^{2}_{N}}\right)-\frac{m^{2}_{S_{2}}}{m^{2}_{S_{2}}-m^{2}_{N}}\ln\left(\frac{m^{2}_{S_{2}}}{m^{2}_{N}}\right)\right]~{},$$

(18)

where we have included $A_{i}$ contributions, $m_{A_{i}}=m_{S_{i}}$ is used, and the symmetric Yukawa couplings $\overline{y}_{ij}$ in flavor indices are defined as:

$$\overline{y}_{ij}=y^{*}_{1i}y^{*}_{2j}+y^{*}_{2i}y^{*}_{1j}~{}.$$

(19)

In the Ma model, the involved Yukawa couplings for $m^{\nu}_{ij}$ appear in the combination of $y_{i}y_{j}$; thus, the mass matrix elements have strong correlations. It needs at least two singlet right-handed fermions to explain the neutrino data. In our model, due to the introduction of one more inert Higgs doublet $\eta_{2}$, the induced neutrino mass matrix elements appear in the combination of $y_{1i}y_{2j}+y_{2i}y_{1j}$ and have less correlations among the matrix elements. It is for this reason that the neutrino data can be accommodated using only one singlet fermion.

The symmetric mass matrix $m^{\nu}$ can be diagonalized through $m^{\nu}_{\rm dia}=U^{\dagger}m^{\nu}U^{*}$, where $m^{\nu}_{\rm dia}={\rm dia}(m_{1},m_{2},m_{3})$ is the diagonal mass matrix. The unitary matrix $U$ can be parametrized as:

$$U=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\rm CP}}\\ -s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta_{\rm CP}}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta_{\rm CP}}&c_{13}s_{23}\\ s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta_{\rm CP}}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta_{\rm CP}}&c_{13}c_{23}\end{array}\right)\left(\begin{array}[]{ccc}e^{i\alpha_{1}}&&\\ &e^{i\alpha_{2}}&\\ &&1\end{array}\right)~{},$$

where $c_{ij}~{}(s_{ij})\equiv\cos\theta_{ij}~{}(\sin\theta_{ij})$, $\delta_{\rm CP}$ is the Dirac CP-violating phase, and $\alpha_{1,2}$ are the Majorana phases. Consequently, the theoretical mass matrix obtained in Eq. (18) can be determined by the observables from the neutrino oscillations, and their relations are given by:

$\displaystyle\begin{split}(m^{\nu})_{11}=&~{}m_{1}c^{2}_{12}c^{2}_{13}e^{i2\alpha_{1}}+m_{2}s^{2}_{12}c^{2}_{13}e^{i2\alpha_{2}}+m_{3}s^{2}_{13}e^{-2i\delta_{\rm CP}}~{},\\ (m^{\nu})_{12}=&~{}m_{1}e^{2i\alpha_{1}}c_{12}c_{13}\left(-s_{12}c_{23}-e^{i\delta_{\rm CP}}c_{12}s_{13}s_{23}\right)\\ &+m_{2}e^{2i\alpha_{2}}s_{12}c_{13}\left(c_{12}c_{23}-e^{i\delta_{\rm CP}}s_{12}s_{13}s_{23}\right)+m_{3}e^{-i\delta_{\rm CP}}s_{13}c_{13}s_{23}~{},\\ (m^{\nu})_{13}=&~{}m_{1}e^{2i\alpha_{1}}c_{12}c_{13}\left(s_{12}s_{23}-e^{i\delta_{\rm CP}}c_{12}s_{13}s_{23}\right)\\ &+m_{2}e^{2i\alpha_{2}}s_{12}c_{13}\left(-c_{12}s_{23}-e^{i\delta_{\rm CP}}s_{12}s_{13}c_{23}\right)+m_{3}e^{-i\delta_{\rm CP}}s_{13}c_{13}c_{23}~{},\\ (m^{\nu})_{22}=&~{}m_{1}e^{2i\alpha_{1}}\left(-s_{12}c_{23}-e^{i\delta_{\rm CP}}c_{12}s_{13}s_{23}\right)^{2}\\ &+m_{2}e^{2i\alpha_{2}}\left(c_{12}c_{23}-e^{i\delta_{\rm CP}}s_{12}s_{13}s_{23}\right)^{2}+m_{3}c^{2}_{13}s^{2}_{23}~{},\\ (m^{\nu})_{23}=&~{}m_{1}e^{2i\alpha_{1}}\left(s_{12}s_{23}-e^{i\delta_{\rm CP}}c_{12}s_{13}c_{23}\right)\left(-s_{12}c_{23}-e^{i\delta_{\rm CP}}c_{12}s_{13}s_{23}\right)\\ &+m_{2}e^{2i\alpha_{2}}\left(-c_{12}s_{23}-e^{i\delta_{\rm CP}}s_{12}s_{13}c_{23}\right)\left(c_{12}c_{23}-e^{i\delta_{\rm CP}}s_{12}s_{13}s_{23}\right)\\ &+m_{3}e^{-i\delta_{\rm CP}}c^{2}_{13}s_{23}c_{23}~{},\\ (m^{\nu})_{33}=&~{}m_{1}e^{2i\alpha_{1}}\left(s_{12}s_{23}-e^{i\delta_{\rm CP}}c_{12}s_{13}c_{23}\right)^{2}\\ &+m_{2}e^{2i\alpha_{2}}\left(-c_{12}s_{23}-e^{i\delta_{\rm CP}}s_{12}s_{13}c_{23}\right)^{2}+m_{3}c^{2}_{13}c^{2}_{23}~{}.\end{split}$

(20)

Among the LFV processes, the most constraining process is the radiative muon decay $\mu\to e\gamma$. It can be the most important process to discover LFV due to the high sensitivity to the new physics effects. In order to study other radiative lepton decays, such as $\tau\to(e,\mu)\gamma$, in the following, we calculate the branching ratios for the $\ell_{i}\to\ell_{j}\gamma$ decays, where $\ell_{j}$ is the lighter lepton and its mass is neglected unless otherwise stated.

The radiative LFV process in the model arises from the $\eta^{\pm}_{i}$-mediated diagrams, as shown in Fig. 2. Using the Yukawa couplings in Eq. (10), the loop-induced effective Lagrangian for $\ell_{i}\to\ell_{j}\gamma^{(*)}$ can be expressed as:

$${\cal L}^{\gamma}_{\ell_{i}\ell_{j}\gamma}=ek^{2}C^{\gamma}_{1ji}\,\overline{\ell}_{j}\gamma_{\mu}P_{L}\ell_{i}A^{\mu}-\frac{em_{\ell_{i}}}{2}C^{\gamma}_{2ji}\,\overline{\ell}_{j}\sigma_{\mu\nu}P_{R}\ell_{i}F^{\mu\nu}~{},$$

(21)

where the loop induced Wilson coefficients are obtained as:

$\displaystyle\begin{split}C^{\gamma}_{1ji}&=\sum^{2}_{\alpha=1}\frac{y^{*}_{\alpha i}y_{\alpha j}}{(4\pi)^{2}m^{2}_{\eta^{\pm}_{\alpha}}}J^{\gamma}_{1}\left(\frac{m^{2}_{N}}{m^{2}_{\eta^{\pm}_{\alpha}}}\right)~{},\\ C^{\gamma}_{2ji}&=\sum^{2}_{\alpha=1}\frac{y^{*}_{\alpha i}y_{\alpha j}}{(4\pi)^{2}m^{2}_{\eta^{\pm}_{\alpha}}}J^{\gamma}_{2}\left(\frac{m^{2}_{N}}{m^{2}_{\eta^{\pm}_{\alpha}}}\right)~{},\end{split}$

(22)

and the loop integral functions are defined by

$\displaystyle\begin{split}J^{\gamma}_{1}(x)&=\frac{1}{36(1-x)^{4}}\left(2-9x+18x^{2}-11x^{3}+6x^{3}\ln x\right)~{},\\ J^{\gamma}_{2}(x)&=\frac{1}{12(1-x)^{4}}\left(1-6x+3x^{2}+2x^{3}-6x^{2}\ln x\right)~{}.\end{split}$

(23)

The emitted photon can be on-shell ($k^{2}=0$) or off-shell ($k^{2}\neq 0$), where the latter can be used to study the $\ell_{i}\to 3\ell_{j}$ process, in which a $\ell_{j}^{+}\ell_{j}^{-}$ pair is produced by the off-shell photon. (The contribution of the $Z$-mediated diagrams is found to be small and will be neglected, as discussed in the next section.) Using the effective Lagrangian in Eq. (21), the branching ratio for $\ell_{i}\to\ell_{j}\gamma$ can be obtained as Toma:2013zsa :

$$BR(\ell_{i}\to\ell_{j}\gamma)=\frac{3(4\pi)^{3}\alpha_{\rm em}}{4G^{2}_{F}}\left|C^{\gamma}_{2ji}\right|^{2}BR(\ell_{i}\to\ell_{j}\overline{\nu}_{j}\nu_{i})~{},$$

(24)

with $\alpha_{\rm em}=e^{2}/(4\pi)$.

In addition to the radiative LFV decays, the dipole operator in Eq. (21) can contribute to the lepton anomalous magnetic dipole moment ($g-2$) when we replace $\ell_{j}$ by $\ell_{i}$. As a result, the lepton $(g-2)$ can be directly obtained as:

$$a_{\ell}\equiv\left(\frac{g-2}{2}\right)_{\ell}=-m^{2}_{\ell}C^{\gamma}_{2\ell\ell}~{}.$$

(25)

We now discuss the three-body lepton flavor-changing decays. For simplicity, we only concentrate on the $\ell_{i}\to 3\ell_{j}$ decay. In the model, the LFV processes $\ell_{i}\to 3\ell_{j}$ can arise from photon-penguin, $Z$-penguin, and box diagrams. In the following, we discuss their contributions in sequence.

Since the effective interaction for $\ell_{i}\to\ell_{j}\gamma^{*}$ has been given in Eq. (21), the transition amplitude with the assigned momenta for the $\ell_{i}(p)\to\ell_{j}(p_{1})\ell^{+}(p_{2})\ell^{-}(p_{3})$ decay can be written as:

$\displaystyle\begin{split}{\cal M}_{\gamma}=&~{}e^{2}C^{\gamma}_{1ji}\overline{u}(p_{1})\gamma_{\mu}u(p)\,\overline{u}(p_{3})\gamma^{\mu}v(p_{2})\\ &+\frac{e^{2}m_{\ell_{i}}}{k^{2}}C^{\gamma}_{2ji}\overline{u}(p_{1})i\sigma_{\mu\nu}k^{\nu}u(p)\,\overline{u}(p_{3})\gamma^{\mu}v(p_{2})-(p_{1}\leftrightarrow p_{3})~{},\end{split}$

(26)

where the photon coupling to the SM charged lepton shown in Eq. (16) is applied.

The Feynman diagrams for $\ell_{i}\to\ell_{j}Z^{*}$ are similar to the photon diagrams shown in Fig. 2, where we use $Z^{*}$ instead of $\gamma^{*}$. Using the $Z$ gauge couplings to $\eta^{\pm}$ and the SM lepton, given in Eqs. (14) and (16), the one-loop induced effective interaction for $\ell_{i}\ell_{j}Z$ can be obtained as:

$\displaystyle{\cal L}^{Z}_{\ell_{i}\ell_{j}Z}$

$\displaystyle=\frac{gc_{2W}}{c_{W}}\sum^{2}_{\alpha=1}\frac{m_{\ell_{i}}m_{\ell_{j}}}{(4\pi)^{2}m^{2}_{\eta_{\alpha}}}J^{\gamma}_{2}\left(\frac{m^{2}_{N}}{m^{2}_{\eta_{\alpha}}}\right)\overline{\ell}_{j}\gamma_{\mu}P_{R}\ell_{i}Z^{\mu}~{},$

(27)

where we have dropped the small contributions from $k^{2}$ and $k$ terms Toma:2013zsa . It can be clearly seen that the loop-induced $Z$ coupling is proportional to the product of the lepton masses, i.e., $m_{\ell_{i}}m_{\ell_{j}}$. We note that because $N$ does not couple to $Z$, the $Z$ boson cannot be emitted from the fermion line in the loop; therefore, no chiral enhancement factor, e.g., $m^{2}_{N}/m^{2}_{\eta^{\pm}_{i}}$, appears in the model. When a necessary mass insertion occurs in the initial lepton $\ell_{i}$, in order to balance the chirality of the lepton vector current that couples to the vector gauge boson $Z$, a mass factor is necessarily inserted in the final lepton $\ell_{j}$. Hence, we get the result proportional to $m_{\ell_{i}}m_{\ell_{j}}$. Due to the fact that $m_{\ell_{j}}\ll m_{\ell_{i}}$ and $m_{\ell_{i}}m_{\ell_{j}}/m^{2}_{Z}\ll 1$, we thus neglect the $Z$-penguin contributions to $\ell_{i}\to 3\ell_{j}$.

The box diagrams contributing to $\ell_{i}\to 3\ell_{j}$ are partly shown in Fig. 3. In addition to the diagrams mediated by the same $\eta^{\pm}_{1(2)}$, the other possible diagrams are mediated by both $\eta^{\pm}_{1}$ and $\eta^{\pm}_{2}$ and involve chirality-flipping factor $m^{2}_{N}$. Using the Yukawa couplings in Eq. (10), the four-fermion interaction for $\ell_{i}\to 3\ell_{j}$ is obtained as:

$${\cal M}_{\rm box}=(C^{\rm box}_{1ji}+C^{\rm box}_{2ji})\,\overline{\ell}_{j}\gamma_{\mu}P_{L}\ell_{j}\,\overline{\ell}_{j}\gamma^{\mu}P_{L}\ell_{i}~{},$$

(28)

where $C^{\rm box}_{1}$ denotes the contributions from diagrams involving the same $\eta^{\pm}_{1(2)}$ and $C^{\rm box}_{2}$ are from those involving both $\eta^{\pm}_{1}$ and $\eta^{\pm}_{2}$, and the effective Wilson coefficients are given by:

$\displaystyle\begin{split}C^{\rm box}_{1ji}&=-\sum^{2}_{\alpha=1}\frac{y^{*}_{\alpha i}y_{\alpha j}|y_{\alpha j}|^{2}}{(4\pi)^{2}m^{2}_{\eta_{\alpha}}}J^{\rm box}_{1}\left(\frac{m^{2}_{N}}{m^{2}_{\eta_{\alpha}}}\right)~{},\\ C^{\rm box}_{2ji}&=\frac{y^{*}_{1i}y_{1j}|y_{2j}|^{2}+y^{*}_{2i}y_{2j}|y_{1j}|^{2}}{(4\pi)^{2}m^{2}_{\eta_{2}}}\frac{m^{2}_{N}}{m^{2}_{\eta_{2}}}J^{\rm box}_{2}\left(\frac{m^{2}_{N}}{m^{2}_{\eta_{2}}},\frac{m^{2}_{\eta_{1}}}{m^{2}_{\eta_{2}}}\right)~{},\end{split}$

(29)

with the loop integral functions given by

$\displaystyle\begin{split}J^{\rm box}_{1}(x)&=\frac{1+x}{2(1-x)^{2}}+\frac{\ln x}{(1-x)^{3}}~{},\\ J^{\rm box}_{2}(x,y)&=-\frac{1}{(1-x)(y-x)}-\frac{(y-x^{2})\ln x}{(1-x)^{2}(y-x)^{2}}+\frac{y\ln y}{(1-y)(y-x)^{2}}~{}.\end{split}$

(30)

Although $C^{\rm box}_{2ji}$ seems to have a chiral enhancement, its contribution is in fact somewhat smaller than $C^{\rm box}_{1ji}$ due to the assumed condition that $m_{N}<m_{\eta_{2}}$.